How does Irving in "The Legend of Sleepy Hollow" make use of American folklore tradition? Identify the gothic, the sublime, and correspondence as Romantic forms.

In the eighteenth century, writers began to explore the sublime. The sublime is characterized by the awe a person feels while observing a natural scene that has a striking beauty or majesty. The feelings evoked by the scene put us in touch, the eighteenth-century school of sensibility and the Romantics believed, with God's presence in the world. Trying to put this feeling into words led Romantic writers to compose long passages of lovely description. The sublime can be found in this story in the following scene that Ichabod sees from afar as he rides along:

The wide bosom of the Tappan Zee lay motionless and glassy, excepting that here and there a gentle undulation waved and prolonged the blue shadow of the distant mountain. A few amber clouds floated in the sky, without a breath of air to move them. The horizon was of a fine golden tint, changing gradually into a pure apple green, and from that into the deep blue of the mid-heaven. A slanting ray lingered on the woody crests of the precipices that overhung some parts of the river, giving greater depth to the dark-gray and purple of their rocky sides. A sloop was loitering in the distance, dropping slowly down with the tide, her sail hanging uselessly against the mast; and as the reflection of the sky gleamed along the still water, it seemed as if the vessel was suspended in the air.

Gothic elements are more prevalent than the sublime in this story, though they are framed in comic tones. The Romantics were interested in the Gothic because it countered the cold rationalism of Enlightenment thinking. The Gothic is characterized by the dark, the irrational, and the uncanny. The uncanny is that which is disturbing, most often represented by the corpse. Ghosts are also emblems of the uncanny. A headless horseman would be uncanny: a headless man must be a corpse, but it is irrational and disturbing that a corpse could ride a horse.
Irving utilizes Gothic imagery throughout the story, although he does so in a lighthearted way that pokes fun at it. I will quote one example, but you can easily find more. This is Ichabod's view of the headless horseman:

Huge, misshapen, black, and towering. It stirred not, but seemed gathered up in the gloom, like some gigantic monster ready to spring upon the traveller.

I am not sure what you mean by "correspondence," but correspondence is a literary device in which a person is made to correspond with an animal or some aspect of nature. In this story, Ichabod corresponds with the ghostly, Gothic imagery of the story. For example, he is always reading about old superstitions, witches, and ghosts. He is thin and associated with "famine descending on the earth" [namely, death and the uncanny]. Both Brom and Katrina, on the other hand, correspond with concrete, material, life-giving images. Katrina, for example, is associated strongly with the solid, material world Ichabod lusts after. At her house he finds,

The doughty dough-nut, the tenderer oly koek, and the crisp and crumbling kruller; sweet-cakes and short-cakes, ginger-cakes and honey-cakes, and the whole family of cakes. And then there were apple-pies, and peach-pies, and pumpkin-pies; besides slices of ham and smoked beef. . .